Accurate calculation of oblate spheroidal wave functions

TL;DR

This paper introduces new methods for accurately computing oblate spheroidal radial functions across large parameter ranges, overcoming limitations of traditional expressions, and provides a Fortran program for practical use.

Contribution

It presents alternative expressions and a modified method for precise calculation of oblate spheroidal radial functions, enabling high accuracy with 64 and 128 bit arithmetic.

Findings

Accurate values for R1ml and R2ml over large parameter ranges.

A Fortran program Oblfcn achieves at least 8-digit accuracy.

Methods extend to large c and l values, improving computational reliability.

Abstract

Alternative expressions for calculating the oblate spheroidal radial functions of both kinds R1ml and R2ml are shown to provide accurate values over very large parameter ranges using 64 bit arithmetic, even where the traditional expressions fail. First is the expansion of the product of a radial function and the angular function of the first kind in a series of products of the corresponding spherical functions, with the angular coordinate being a free parameter. Setting the angular coordinate equal to zero leads to accurate values for R2ml when the radial coordinate xi is larger than 0.01 and l is somewhat larger than m. Allowing it to vary with increasing l leads to highly accurate values for R1ml over all parameter ranges. Next is the calculation of R2ml as an integral of the product of S1ml and a spherical Neumann function kernel. This is useful for smaller values of xi. Also used is the near equality of pairs of low order eigenvalues when the size parameter c is large that leads to accurate values for R2ml using neighboring accurate values for R1ml. A modified method is described that provides accurate values for the necessary expansion coefficients when c is large and l is near m and traditional methods fail. A resulting Fortran computer program Oblfcn almost always provides radial function values with at least 8 accurate decimal digits using 64 bit arithmetic for m up to at least 1000 with c up to at least 2000 when xi is greater than 0.000001 and c up to at least 5000 when xi is greater than 0.01. Use of 128 bit arithmetic extends the accuracy to 15 or more digits and extends xi to all values other than zero. Oblfcn is freely available.